List of practice Questions

Let f : R $\to$ R be the Signum Function defined as $f(x) = \begin{cases} 1, & \quad \text x>0 \\ 0, & \quad x=0 \\ -1, &\quad x<0 \end{cases}$

and $g: R \to R$ be the Greatest Integer Function given by g (x)= [x], where [x] is greatest integer less than or equal to x. Then does fog and gof coincide in (0,1]?

The pH value of N/10 NaOH is

Interfascicular cambium and cork cambium are formed due to

Define a binary operation *on the set {0,1,2,3,4,5} as
$a*b = \begin{cases} a+b & \quad \text{if } a+b<6 \\ a+b-6 & \quad \text{if } a+b\geq6 \end{cases}$
Show that zero is the identity for this operation and each element a≠0 of the set is invertible with 6−a being the inverse of a.

Given a non-empty set X, let *:P (X)×P (X)$\to$ P (X) be defined as A * B= (A−B)∪(B−A),∀ A,B∈ P (X).
Show that the empty set $\Phi$ is the identity for the operation * and all the elements A of P (X) are invertible with $A^{-1}=A.$
(Hint: $(A-\Phi)\cup(\Phi-A)=A\,and\,(A-A)\cup(A-A)=A*A=\Phi)$.

Consider the binary operations*: R ×R → and o: R×R →R defined as $a*b=\mid a-b \mid $and a o b = a,∀a,b∈R.
Show that * is commutative but not associative, o is associative but not commutative.
Further, show that ∀a,b,c∈R, a * (b o c)= (a * b) o (a * c). [If it is so, we say that the operation * distributes over the operation o]. Does o distribute over *? Justify your answer.

Which of the following gives maximum energy in metabolic process?

$[Fe(CN)_6 ]^{3-}$ ion is

Let A={1,2,3}. Then number of relations containing (1,2) and (1,3) which are reflexive and symmetric but not transitive is

Let A={1,2,3}. Then number of equivalence relations containing (1,2) is

Let S = {a,b,c} and T= {1,2,3}.
Find $F^{-1}$ of the following functions F from S to T, if it exists.
I. F={(a,3),(b,2),(c,1)}
II. F={(a,2),(b,1),(c,1)}

Given a non empty set X, consider P(X) which is the set of all subsets of X.
Define the relation R in P(X) as follows:
For subsets A,B in P(X),ARB if and only if A⊂B.
Is R an equivalence relation on P(X)? Justify you answer:

Given a non-empty set X, consider the binary operation * : P (X)×P (X)→P (X) given by A * B= A∩B ∀A,B in P (X) is the power set of X. Show that X is the identity element for this operation and X is the only invertible element in P (X) with respect to the operation*.

If $a =\hat{ i }+2 \hat{ j }+2 \hat{ k },| b |=5$ and the angle between $a$ and $b$ is $\pi / 6$, then the area of the triangle formed by these two vectors as two sides is

If $f:R\to R$ is defined by $f(x)=x^2-3x+2,find \,f(f(x)).$

Let f : W $\to$ W be defined as f(n)=n−1, if is odd and f(n)=n+1, if n is even. Show that f is invertible. Find the inverse of f. Here, W is the set of all whole numbers.

$3sin^{-1}(3x-4x^3), x \in[-\frac {1}{2},\frac{1}{2}]$ prove.

Let f : R →R be defined as $f(x)=10x+7.$ Find the function g : f→R such that gof=f o g=1_R.

For each binary operation * defined below, determine whether * is commutative or associative.
(i) On Z, define a * b=a−b
(ii) On Q, define a * b=ab+1
(iii) On Q, define a * b= $\frac {ab}{2}$.
(iv) On Z⁺, define a * b=2ab
(v) On Z⁺, define a * b=ab
(vi) On R−{−1},define a * b= $\frac {a}{b+1}$

Show that the relation R in the set A of all the books in a library of a college, given by R = {(x, y): x and y have same number of pages} is an equivalence relation.

Find the value of $tan^{-1\sqrt3-sec^{-1}(-2)}$ is equal to

Find the value of if sin^-1$x=y$, then

What is the perpendicular distance of the point P(6, 7, 8) from xy-plane?

Pathogenecity of bacteria causing tuberculosis and leprosy is due to

Find the principal value of cosec^-1$(-\sqrt2)$